How can I export a model with its textures applied?
The most likely explanations for why your textures are not exporting would be that (a) the textures are not in an image format supported by the export format, or (b) the material is not configured in a way that the export format's addon understands.
In the case of glTF 2.0 (I can't comment on FBX or OBJ), images must be in PNG or JPG format to be exported. The material should typically use the Principled BSDF material — arbitrary material node graphs cannot be exported from Blender to any format.
For full details on how to configure your materials for correct export from Blender, see the documentation of the Blender glTF exporter. In particular, this is a standard material setup with a base color texture: